A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 116 inches.

Question

A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 116 inches. Find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements.

Maximum Volume ____ in^3

Answer 1

Length plus girth does not exceed 116 inches

L+2h+2w < 116

L = 116-2h-2w

square ends mean h=w

L= 116-4w

Volume = Length times width times height= Lw^2

v =Lhw = (116-4w)w^2

v = 116w^2 -4w^3

v'(w) = 232w-12w^2 = 0

232 = 12w

w = 232/12 = 19 1/3 inches wide and high

Length =116-4(232/12) = 116-232/3 = 348/3-232/3

= 116/3 = 38 2/3 inches Long

19 1/3 in by 19 1/3 in by 38 2/3 inches

twice as Long as it's high or wide

max v = (19 1/3)^2(38 2/3)=

= about 14,452.74074... cubic inches